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Geometric Representation Theory and Beyond

There are powerful new tools and ideas at the forefront of geometric representation theory, particularly categorical incarnations of ideas from mathematical physics, algebraic, arithmetic and symplectic geometry, and topology: perhaps geometry is no longer the future of geometric representation theory. Speakers at this workshop have been invited to present their own vision for the future of the field.

Speakers: David Ben-Zvi (Austin), Alexander Braverman (Toronto and Perimeter Institute for Theoretical Physics), Laurent Fargues (Jussieu), Dennis Gaitsgory (Harvard), Victor Ginzburg (Chicago), Michael Harris (Columbia), Tamas Hausel (IST Austria), David Nadler (Berkeley), Andrei Okounkov (Columbia), Vera Serganova (Berkeley), Wolfgang Soergel (Freiburg), Catharina Stroppel (Bonn), Constantin Teleman (Oxford), Geordie Williamson (Bonn)

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Venue: Mathematical Institute, University of Oxford

Mean Curvature Flow

Mean curvature flow (MCF) is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time.  It has been studied in material science to model things such as cell, grain, and bubble growth.  It also plays an important role in image processing and has been widely studied numerically.   The flow is well-defined classically until the first singularity.  The level set method was invented to study the flow, even past singularities, and it is now widely used in a variety of settings.

Before the flow becomes extinct, changes occur as it goes through singularities.   To understand the flow, it is crucial to understand these.  This can be done completely for embedded curves, but there are infinitely many possible types of singularities in all higher dimensions.  Recent years have seen major progress on some of the most natural questions:  Is it possible to bring the hypersurface into general position so that there is a smaller set of “generic” singularities?  What does the flow look like near these singularities?  What is the structure of the singular set itself? 

This workshop will explore recent developments in MCF, focusing on which singularities can occur, which of these are generic, what the flow looks like near these singularities, and the structure of the singular set.

Speakers: Sigurd Angenent (Wisconsin), Jacob Bernstein (Johns Hopkins), Yoshikazu Giga (Tokyo), Gerhard Huisken (Tübingen), Bruce Kleiner (NYU), Bob Kohn (NYU), Felix Otto (MPIM Leipzig), Felix Schulze (UCL), James Sethian (Berkeley), Peter Topping (Warwick), Lu Wang (Wisconsin), Mu-Tao Wang (Columbia), Brian White (Stanford)

Some of the topics that will be covered in the workshop are described in the paper Level set flow  by Colding and Minicozzi.

Image:  R. I. Saye and J. A. Sethian, Lawrence Berkeley National Laboratory and University of California, Berkeley

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Venue: Mathematical Institute, University of Oxford

Recent Developments on Elliptic Curves

The last few years have witnessed a number of developments in the arithmetic of elliptic curves, notably the proof that there are positive proportions of elliptic curves of rank zero and rank one for which the Birch—Swinnerton-Dyer conjecture is true.  The proof of this landmark result relies on an appealing mix of diverse techniques arising from the newly resurgent field of arithmetic invariant theory, Iwasara theory, congruences between modular forms, and the theory of Heegner points and related Euler systems.  The purpose of this workshop is to survey the proof of this theorem and to describe the new perspectives on the Birch—Swinnterton-Dyer conjecture which it opens up.

Speakers: Mirela Ciperiani (Texas, Austin), Ellen Eischen (Oregon), Benedict Gross (Harvard), Wei Ho (Michigan), Antonio Lei (Laval), Chao Li (Columbia), Kartik Prasanna (Michigan), Victor Rotger (Catalunya), Arul Shankar (Harvard), Ye Tian ( Chinese Academy of Sciences), Eric Urban (Columbia), Rodolfo Venerucci (Duisberg-Essen), Xin Wan (Columbia), Xiaoheng Jerry Wang (Princeton), Andrew Wiles (Oxford), Shou-Wu Zhang (Princeton)

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Venue: Mathematical Institute, University of Oxford

Modern Moduli Theory

The workshop will focus on modern approaches to moduli problems in algebraic geometry, including derived structures on moduli spaces such as shifted symplectic and shifted Poisson structures, novel quotient constructions, and relationships to geometric representation theory.

Speakers: Dario Beraldo (Oxford), Chris Brav (HSE, Moscow), Yalong Cao (Oxford), Ben Davison (IST Vienna), Chris Dodd (Illinois), Barbara Fantechi (SISSA Trieste), Elham Izadi (San Diego), Frances Kirwan (Oxford), Kobi Kremnitzer (Oxford), Sven Meinhardt (Sheffield), Tom Nevins (Illinois), Georg Oberdieck (MIT), Andrei Okounkov (Columbia), Tony Pantev (Penn), Jørgen Rennemo (Oslo and Oxford), Nick Rozenblyum (Chicago), Yukinobu Toda (IPMU, Tokyo), Bertrand Toën (Toulouse)

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Venue: Mathematical Institute, University of Oxford

Nonlocal PDEs

This workshop will focus on recent development in the area of nonlocal PDEs and its various applications.  It will bring together experts from different fields, including analysis, geometry, numerical analysis, probability, fluid dynamics, and others.

Speakers: Xavier Cabre (Catalunya), María del Mar Gonzáles (Madrid), Cyril Imbert (ENS Paris), Tianling Jin (HKUST), Moritz Kassmann (Bielefeld), Dennis Kriventsov (NYU), Jean-Michel Roquejoffre (Toulouse), Joaquim Serra (ETH), Yannick Sire (Johns Hopkins), Enrico Valdinoci (Melbourne), Alexis Vasseur (Austin), Juan Luis Vazquez (Madrid), Jun-cheng Wei (British Columbia)

Image © Walter Baxter, Wikimedia Commons

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Venue: Mathematical Institute, University of Oxford

Ergodic Theory: Numbers, Fractals, and Geometry

Speakers: Nalini Anantharaman (Strasbourg), Tim Austin (UCLA), Lewis Bowen (Austin), Emmanuel Breuillard (Paris Sud), Giovanni Forni (Maryland), Alex Gorodnik (Bristol), Alex Kontorovich (Rutgers), Elon Lindenstrauss (HUJ), Jens Marklof (Bristol), Hee Oh (Yale), Jean-François Quint (Bordeaux), Pablo Shmerkin (Torcuata Di Tella), Corinna Ulcigrai (Bristol), Péter Varjú (Cambridge)

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Venue: Mathematical Institute, University of Oxford

Harmonic Analysis and Related Areas

This workshop will explore harmonic analysis and its connections to other areas, including combinatorics, analytic number theory, and ergodic theory.  Talks will discuss recent progress, and may introduce people in one field to ideas and problems from another related field.

Speakers:  Nalini Anantharaman (Strasbourg), Jonathan Bennett (Birmingham), Ciprian Demeter (Indiana), Polona Durcik (Caltech), Ben Green (Oxford), Izabella Laba (British Columbia), Elon Lindenstrauss (HUJ), Stefanie Petermichl (Toulouse III), Tom Sanders (Oxford), Christopher Sogge (Johns Hopkins), Christoph Thiele (Bonn), Trevor Wooley (Bristol), Joshua Zahl (British Columbia)

Image:  Gtgith, Wikimedia Commons

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Venue: Mathematical Institute, University of Oxford

Discrete Structures: Harmonic Analysis and Probability

This workshop will focus  on progress in analysis of geometric functional inequalities in particular, but not exclusively, on discrete model: the Hamming cube.

Talks will give an overview of the area, its connections to PDE, combinatorics, martingale methods, stochastic optimization problems, various notions of convexity, and applications to random graphs.  The harmonic analysis on Hamming cube is a tool for studying the Renyi random graph model (e.g. sharp threshold results of the type of Margulis graph connectivity theorem). 

The geometric functional inequalities are often rooted in their discrete versions on Hamming cube, where the harmonic analysis has many interesting specifics and often uses probabilistic ideas. For example, martingale method leads to important geometric inequalities on Hamming cube and  Gauss space, such as e.g. the concentration of measure inequalities. The concentration of measure phenomenon, in its turn, has utmost importance in applied mathematics, in questions of spectral initialization or compressed sensing.

Speakers:  Franck Barthe (Toulouse), Sergey Bobkov (Minnesota), Dario Cordero-Erausquin (Paris 6), Paata Ivanisvili (UC Irvine), Ryan O’Donnell (Carnegie Mellon), Stefanie Petermichl (Toulouse), Mark Rudelson (Michigan), Ramon van Handel (Princeton), Alexander Volberg (Michigan State)

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Venue: Mathematical Institute, University of Oxford

Analysis and Probability

This workshop will focus on the recent progress in Analysis with particular attention to the problems that are either directly motivated by Probability or use probabilistic ideas and techniques. For example, recent progress of Logunov and Malinnikova who proved the Nadirashvili’s conjecture and obtained the lower bound in Yau’s conjecture about the size of nodal sets for Laplace eigenfunctions or analytic proof by Tolsa and Volberg of Tsirelson’s theorem about harmonic measures. Talks will give an overview of the current state of the area and present recent progress in several long-standing open questions.

Participants: Kari Astala (Aalto), Oleg Ivrii (Caltech), Nikolai Makarov (Caltech), Andrei Okounkov (Columbia), Alexei Poltoratski (Texas A&M), Eero Saksman (Helsinki), Misha Sodin (Tel Aviv), Xavier Tolsa (UA Barcelona), Yilin Wang (ETH)

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Venue: Mathematical Institute, University of Oxford

Modular Representation Theory

Modular representation theory is the study of representations of groups and related algebras over fields of positive characteristic. It was initially developed by Brauer, with a view towards the structure of finite groups. Over the last decades it has grown into a deep and beautiful subject with connections to number theory, the Langlands program and algebraic geometry.

The goal of this workshop is to explore the breadth of modern modular reprensentation theory. There will be talks from relevant areas including: mod p Langlands correspondence, deformations of Galois representations, modular representations of algebraic groups, Lie algebras and p-adic groups, modular Deligne-Lusztig theory, and local-global conjectures in finite groups.

Speakers:  Noriyuki Abe (Tokyo), Pramod Achar (LSU), Roman Bezrukavnikov (MIT), Cédric Bonnafé (Montpellier), Ana Caraiani (Imperial), Joe Chuang (City London), Olivier Dudas (Paris 7), Jens Eberhardt (UCLA), Amit Hazi (City London), Bao Le Hung (Northwestern), Brandon Levin (Arizona), Carl Mautner (UC Riverside), Gil Moss (Utah), Gabriel Navarro (Valencia), Simon Riche (Clermont Auvergne), Raphael Rouquier (UCLA), Peng Shan (Tsinghua), Jack Thorne (Cambridge)

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Venue: Mathematical Institute, University of Oxford

Periods, Representations, and Arithmetic: Recent Advances on the Gan-Gross-Prasad Conjectures and their Applications

In the early 2000s, Gan, Gross, and Prasad  formulated a series of conjectures connecting the representation theory of classical groups, periods of automorphic forms, algebraic cycles on Shimura varieties, and special values of L-functions. There has been a lot of progress since on their conjectures and the arithmetic applications. This workshop will explore recent developments as well as possible future directions in the subject. 

Speakers:  Raphaël Beuzart-Plessis (Marseille), Pierre-Henri Chaudouard (Paris 7), Dimitar Jetchev (EPFL), Rahul Krishna (Northwestern), Chao Li (Columbia), Philippe Michel (EPFL), Andreas Mihatsch (MIT), Dipendra Prasad (TIFR), Kartik Prasanna (Michigan), Michael Rapoport (Bonn), Brian Smithling (Johns Hopkins), Yichao Tian (Strasbourg), Eric Urban (Columbia), Jingwei Xiao (MIT), Hang Xue (Arizona), Shouwu Zhang (Princeton)

Image: Spherical harmonics, inigo.quilez, Wikimedia Commons

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Venue: Mathematical Institute, University of Oxford

Beyond Spectral Gaps

The goal of this workshop is to bring to Oxford leading experts in the various areas of mathematics and theoretical computer science with an interest in the spectral analysis of graphs and its many applications within mathematics. Expander graphs have now become a ubiquitous tool in much of theoretical computer science and attention has focused in the last decade or so, on the one hand, on applications of these methods to other, seemingly unrelated questions both within pure mathematics and within combinatorics and computer science, and on the other hand, to the development of generalizations of the idea of spectral gap. Our goal is to focus on those exciting recent developments where the goal has not been simply to establish a spectral gap in any given situation, but to obtain more refined spectral information with it or from it. Topics will include interlacing polynomials, strong property (T), monotone expanders, local spectral gaps, small-set expansion, affine sieve, non-linear spectral gaps, matrix expanders, high-dimensional expanders, etc.

This workshop is one of the two events composing Pansu’s Fest.  It will be preceded by the conference Geometry and Analysis: Celebrating the Mathematics of Pierre Pansu.  Participants in either of the events are welcome to attend the other event as well.

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Venue: Mathematical Institute, University of Oxford